#10143276

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#51932	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,23,25)(19,20,28,24)(21,26,32,30)(22,29,31,27)(33,34,40,44)(35,41,46,48)(36,42,37,39)(38,43,45,47);(1,3,7,4)(2,8,9,10)(5,12,15,13)(6,11,16,14)(17,19,23,28)(18,24,25,20)(21,31,32,22)(26,29,30,27)(33,35,40,46)(34,41,44,48)(36,47,37,43)(38,39,45,42);(2,9)(3,4)(11,14)(12,13)(18,25)(20,24)(26,30)(27,29)(33,36,40,37)(34,42,44,39)(35,43,46,47)(38,41,45,48);(1,4,7,3)(2,10,9,8)(5,13,15,12)(6,14,16,11)(17,20)(18,19)(21,27)(22,30)(23,24)(25,28)(26,31)(29,32)(33,37,40,36)(34,39,44,42)(35,43,46,47)(38,41,45,48);(1,5,7,15)(2,11,9,14)(3,13,4,12)(6,10,16,8)(17,21,23,32)(18,26,25,30)(19,22,28,31)(20,29,24,27)(33,38)(34,43)(35,39)(36,41)(37,48)(40,45)(42,46)(44,47);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,8,15,10)(17,22)(18,27)(19,21)(20,30)(23,31)(24,26)(25,29)(28,32)(33,39)(34,36)(35,45)(37,44)(38,46)(40,42)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050472);